| L(s) = 1 | + (1.66 − 1.85i)2-s + (−0.651 + 0.290i)3-s + (−0.439 − 4.18i)4-s + (−2.15 + 0.458i)5-s + (−0.549 + 1.69i)6-s + (−2.30 − 1.29i)7-s + (−4.44 − 3.22i)8-s + (−1.66 + 1.85i)9-s + (−2.74 + 4.75i)10-s + (1.5 + 2.59i)12-s + (1.01 + 3.12i)13-s + (−6.24 + 2.10i)14-s + (1.27 − 0.924i)15-s + (−5.15 + 1.09i)16-s + (−0.997 − 1.10i)17-s + (0.648 + 6.17i)18-s + ⋯ |
| L(s) = 1 | + (1.17 − 1.30i)2-s + (−0.376 + 0.167i)3-s + (−0.219 − 2.09i)4-s + (−0.964 + 0.204i)5-s + (−0.224 + 0.690i)6-s + (−0.871 − 0.490i)7-s + (−1.57 − 1.14i)8-s + (−0.555 + 0.617i)9-s + (−0.868 + 1.50i)10-s + (0.433 + 0.750i)12-s + (0.281 + 0.866i)13-s + (−1.66 + 0.562i)14-s + (0.328 − 0.238i)15-s + (−1.28 + 0.273i)16-s + (−0.241 − 0.268i)17-s + (0.152 + 1.45i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0867 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0867 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0125346 + 0.0114900i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0125346 + 0.0114900i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.30 + 1.29i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (-1.66 + 1.85i)T + (-0.209 - 1.98i)T^{2} \) |
| 3 | \( 1 + (0.651 - 0.290i)T + (2.00 - 2.22i)T^{2} \) |
| 5 | \( 1 + (2.15 - 0.458i)T + (4.56 - 2.03i)T^{2} \) |
| 13 | \( 1 + (-1.01 - 3.12i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.997 + 1.10i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (0.723 - 6.88i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (3.24 + 5.62i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.33 - 0.969i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.29 - 0.488i)T + (28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (5.07 + 2.25i)T + (24.7 + 27.4i)T^{2} \) |
| 41 | \( 1 + (9.10 + 6.61i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 5.26T + 43T^{2} \) |
| 47 | \( 1 + (0.155 - 1.48i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (0.298 + 0.0633i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (-1.32 - 12.5i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (12.6 - 2.69i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (-2.28 + 3.96i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.50 + 10.7i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.895 + 8.52i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (3.09 - 3.44i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-0.598 + 1.84i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (1.60 + 2.77i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.574 - 1.76i)T + (-78.4 + 57.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.27049862714552511843720384654, −8.938276742126451472728153273362, −7.80390035397929233048751956831, −6.58441647094891795625536705857, −5.78497717749745799817663587806, −4.64859036300407309220564023337, −3.92459510576989671987992337182, −3.22746823897585982675124209106, −1.93246497109356045345812268350, −0.00556492014542108568945585497,
3.09103416825877966730016583775, 3.68713807898382317178520441288, 4.86685469357920861298414817566, 5.67267646437415648228584828901, 6.44254082760861494977308329456, 7.06520263372327298364797249493, 8.124614538146493343569421022503, 8.669006381505279382914356299730, 9.829375686751550755446740774859, 11.33393248875170882249956406615
